Quantum theory of multiple-input–multiple-output Markovian feedback with diffusive measurements

Abstract

Feedback control engineers have been interested in MIMO (multiple-input multiple-output) extensions of SISO (single-input single-output) results of various kinds due to its rich mathematical structure and practical applications. An outstanding problem in quantum feedback control is the extension of the SISO theory of Markovian feedback by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)] to multiple inputs and multiple outputs. Here we generalize the SISO homodynemediated feedback theory to allow for multiple inputs, multiple outputs, and arbitrary diffusive quantum measurements. We thus obtain a MIMO framework which resembles the SISO theory and whose additional mathematical structure is highlighted by the extensive use of vector-operator algebra.

Quantum filtering for a two-level atom driven by two counter-propagating photons

“…In quantum optics, quantum filters, also known as quantum trajectories or stochastic master equations, are of great importance in measurement feedback control [50,52,23,53]. The problems of quantum filtering for systems driven by Gaussian states, including vacuum state, thermal state, coherent state, and squeezed state, have been well studied, see, e.g., [17,12,34,18]. On the other hand, as single-and multi-photon states can nowadays be generated in real experiments [39,49,51,31,29,36,32,47], more and more research has been concentrated on atomic excitation [49,4,43,40] and quantum filter design [28,26,45,14,20,3] for systems driven by single or multiple photons.…”

Exponential stabilization of quantum systems under continuous non-demolition measurements

“…• The capability of performing several quantum measurements and of applying different unitary feedback controls on a single quantum system motivates the study of multi-input multioutput (MIMO) quantum feedback schemes. A MIMO version of static output feedback was introduced in [14] and an implementation was made in [11]. In addition, we have presented, in the context of quantum error correction [13], a MIMO scheme of this noise-assisted feedback to stabilize a manifold of quantum states.…”

“…It remains to check that it is a supermartingale with exponential decay. By Itō's formula (14), the variable ξ k := √ p k satisfies…”

Quantum multi-channel decoupling

“…One of the significant goals in quantum control is to construct quantum networks for quantum communication where the control of MIMO quantum systems plays an important role [3], [10]. In this work, we introduce a novel concept of using the quantum feedback to decouple multiple quantum channels.…”